The Find the difference quotient of \( f(x)=x^{2}+1 \); that is find \( \frac{f(x+h)-f(x)}{h}, h \neq 0 \). Be sure to simplify.
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To find the difference quotient of \( f(x) = x^2 + 1 \), we first compute \( f(x+h) \): \[ f(x+h) = (x+h)^2 + 1 = x^2 + 2xh + h^2 + 1 \] Now, we can substitute \( f(x+h) \) and \( f(x) \) into the difference quotient: \[ \frac{f(x+h) - f(x)}{h} = \frac{(x^2 + 2xh + h^2 + 1) - (x^2 + 1)}{h} \] This simplifies to: \[ \frac{2xh + h^2}{h} \] Next, we can factor out \( h \): \[ = \frac{h(2x + h)}{h} \] Since \( h \neq 0 \), we can cancel \( h \): \[ = 2x + h \] Thus, the simplified difference quotient is \( 2x + h \).